The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got $k=8\sqrt{x}(x+9)$. The homework is looking for a numerical answer. I have looked through Google and this site, and the closest I've come to an answer is this question and this one.
This is a hw question, but I'm not looking for an answer. I only want an answer if a general formula isn't possible for this type of problem or if plugging in random $k$s is the best solution, because spending 30 minutes on this on a test is not a possibility for me.
Funny thing is this problem does not require calculus. Recall that a tangent line must intersect the function. Setting the equations equal \begin{align} k\sqrt{x} &= 4x + 36\\ k^2x &= 16x + 288x + 1296\\ 0 &= 16x + (288 - k^2)x + 1296 \end{align} Now for any value of $k$, the solution to this equation gives us the $x$ value of intersection. But recall the properties of quadratics, they can have either 2, 1 or no solutions. We know that we can't have two solutions, as that would contradict the fact that the line is a tangent. We know we must have at least one solution. Luckily when a quadratic function has only 1 solution, its discriminant is $0$. Setting the discriminant to $0$ yields \begin{align} (288 - k^2)^2 - 4(16)(1296) &= 0 \\ (288 - k^2)^2 &= 4(16)(1296)\\ 288 - k^2 &= \pm288\\ k = 0 \text{ or } k =24 \end{align} Now we have two values of $k$. The reason there are two is because when solving radical equations (like $5\sqrt{x} + x = 7$) external solutions can be introduced. If $k = 0$ then no intersection will occur.This means the solution must be $k = 24$. This external solution is caused by the fact that the squared version of our function is defined for positive and negative values, while the same is not true for the function itself. This introduces an extra solution.
Be careful though. This reasoning works because a positively sloped-line will always grow faster than a function of a square root. That means if the square root suddenly passes it, the line will eventually catch up and make a second intersection point. The only exception is the tangent. For functions like $\sin x$ this is not the case. As you can think of many positively sloped lines that intersect the curve only once besides the tangent.